APPENDIX A

If we disregard absorption and transmission losses, and consider the asymptotic approximation (where the ray theory holds), the integral of the energy flux over a wavefront inside a Snell beam, is a constant of propagation. This condition can be expressed by  

$$\int_S \rho v A^2 \, ds = \mbox{CONSTANT}$$ (2)

(Aki and Richards, 1980), where S is the wavefront surface, $\rho$ is the density of the medium, and v and A are the velocity and amplitude of the wave. equation (2) applies to both the downward and upward ring-shaped propagating wavefront of Figure , but the constants are different before and after the reflection.

An azimuthly symmetric conical beam is reflected by an interface. Under the assumption of a plane-layered earth, the amplitude at any point of the ring-shaped wavefront depends only on the Snell parameter p and the offset x.

Let's first apply equation (2) for the downgoing beam, at a unit distance from the source, and just before it hits the reflector:

 

$$\int_{S_i} \mbox{sgn}(A_i)A^2_i(p) \, v_z \, ds_i = \int_{S_0} \mbox{sgn}(A_0) A^2_0(p) \, v_0 \, ds_0,$$ (3)

where the indexes and i apply to the wavefronts near the source and near the reflector, respectively. The sgn function introduced in equation (3) does not affect the equality. It is possible to integrate separately infinitesimal beams with positive and negative amplitudes, and the equality will still hold for both independently, since no sign change is possible inside a beam.

Applying equation (2) for the upcoming beam; just after the reflection and just before it hits the surface, we find that  

$$\int_{S_r} \mbox{sgn}(A_r) A^2_r(p) \, v_z \, ds_r = \int_{S_u} \mbox{sgn}(A_u) A^2_u(p) \, v_0 \, ds_u,$$ (4)

where the indexes r and u apply for the wavefronts near the reflector and near the surface, respectively.

I define now an average amplitude square over a wavefront, in such a way that it keeps the ``average sign of the wavefront"  

$$\bar{A^2} = {\int_{S} \mbox{sgn}(A) A^2 ds \over S}.$$ (5)

It is possible then to have an estimation of the average reflection coefficient over the region illuminated by the beam through the following relation  

$$\bar{R} = \mbox{sgn}({\bar{A^2_r}\over \bar{A^2_i}}) \sqrt{ ... ...t {\bar{A^2_r}\over \bar{A^2_i}} \right \Vert {S_r \over S_i}}.$$ (6)

Dividing equation (4) by equation (3) and using equations (6) and (5) results in  

$$\bar{R} = \mbox{sgn}({\bar{A^2_r}\over \bar{A^2_i}}) \sqrt{ ... ... \pi \bar{A^2_0}(p) (\cos\theta_b-\cos\theta_t)} \right \Vert},$$ (7)

where $\bar{A^2_0}(p)$ is an average (in the sense of equation (5)) estimation of the source radiation pattern, for the beam with Snell parameter p, at unit distance from the source. Indexes b and t apply for the bottom and top rays of the beam as illustrated in Figure .

Vertical cross section of the upcoming beam. If we approximate the surface S by a conical surface, the area element ds is given by $ds = 2 \pi x \cos\theta_t dx$.

A last necessary assumption is that  

$$\mbox{sgn}({\bar{A^2_u}\over \bar{A^2_0}}) = \mbox{sgn}({\bar{A^2_r}\over \bar{A^2_i}}),$$ (8)

which is equivalent to the assumption that the signal of the average described by equation (5) does not change during propagation, unless the beam is reflected.

Finally, if the wavefront ring reaching the surface is approximated by a conical surface, the application of straightforward geometry leads to

$$ds_u = 2 \pi x \cos\theta_t dx.$$

Using this result and substituting equation (8) into equation (7), a final expression for the output sample $\bar{R}(p,t_0)$ is found:

 

$$\bar{R}(p,t_0) = H(p_t,p_b,t_0) \mbox{sgn}(\bar{A^2_u}) \sqr... ...nt_{x_b}^{x_t} \mbox{sgn}(A_u) A^2_u(x,t[x,t_0]) x dx } \Vert},$$ (9)

where

 

$$H(p_t,p_b,t_0) = \mbox{sgn}(\bar{A^2_0}) \sqrt{{ \cos\theta_t \over(\cos\theta_b-\cos\theta_t)}}.$$ (10)

All the cosine factors in equation (10) can be expressed in terms of p_t or p_b, the horizontal slownesses of the boundary rays of the beam, according to

$$\cos \theta = \sqrt{ 1 - (v_0 p)^2}.$$

The offsets x_t and x_b that appear in the integral limits of equation (9) are determined by the interceptions (in the x-t domain) of the top and bottom rays with the reflection curve (or wavefront) t=t(x,t₀). The specific function that describes the wavefronts in the x-t domain can be given either explicitly by

$$t = \sqrt{t^2_0 + (x/v(t_0))^2},$$

if an hyperbolic approximation is satisfactory, or implicitly with the use of raytracing.